A fundamental theorem in Elliptic Clifford Analysis (ECA), with the standard vector Dirac operator, is presented that is valid for Clifford algebra-valued distributions. This theorem holds under fairly general conditions on the allowed singularities of the right-hand side distributions and on the region of integration. Next a specialization of this fundamental theorem is proved that forms the starting point for solving boundary value problems with distributional sources in ECA. Finally, distributional equivalents of the Residue theorem, Cauchy's theorem and Cauchy's integral theorem are stated.